Observability of Systems under Uncertainty

Abstract

The authors observe the evolution of a state of a system under uncertainty governed by a differential inclusion through an observation map. The set-valued character due to uncertainty leads them to introduce the "Sharp Input-Output Map", which is a (usual) product, and the "Hazy Input-Output Map", which is a square product. They provide criteria for both sharp and hazy local observability in terms of (global) sharp and hazy observability of a variational inclusion.

They reach their conclusions by implementing the following strategy: (1) Provide a general principle of local injectivity and observability of a set-valued map I, which derives these properties from the fact that the kernel of an adequate derivative of I is equal to 0. (2) Supply chain rule formulas which allow to compute the derivatives of the usual product I_{-} and the square product I_{+} from the derivatives of the observation map H and the solution map S. (3) Characterize the various derivatives of the solution map S in terms of the solution maps of the associated variational inclusions. (4) Piece together these results for deriving local sharp and hazy observability of the original system from sharp and hazy observability of the variational inclusions. (5) Study global sharp and hazy observability of the variational inclusions.